If \(X\)  is complete and supports a \((1,p)\) -Poincaré inequality for some \(1<p<\infty \) , then there exists \(1<s<p\)  such that
\(X\)  supports a \((1,s)\) -Poincaré inequality; see [1]}. Since we do not require the completeness from \(X\)  in
this paper, we can not refer to this self-improvement result and
hence in some of our results we directly assume that \(X\)  supports a \((1,s)\) -Poincaré inequality for some
\(1<s<p\) .
